*Preprint*

**Inserted:** 16 may 2024

**Last Updated:** 30 may 2024

**Year:** 2024

**Abstract:**

We show a sharp and rigid spectral generalization of the classical Bishop-Gromov volume comparison theorem: if a closed Riemannian manifold $(M,g)$ of dimension $n\geqslant 3$ satisfies \[ \lambda_1\left(-\frac{n-1}{n-2}\Delta+\mathrm{Ric}\right)\geqslant n-1, \] then $\operatorname{vol}(M)\leq\operatorname{vol}(\mathbb S^{n})$, and $\pi_1(M)$ is finite. Moreover, the constant $\frac{n-1}{n-2}$ cannot be improved, and if $\mathrm{vol}(M)=\mathrm{vol}(\mathbb S^n)$ holds, then $M\cong \mathbb S^{n}$. A sharp generalization of the Bonnet--Myers theorem is also shown under the same spectral condition. The proofs involve the use of a new unequally weighted isoperimetric problem, and unequally warped $\mu$-bubbles. As an application, in dimensions $3\leqslant n\leqslant 5$, we infer sharp results on the isoperimetric structure at infinity of complete manifolds with nonnegative Ricci curvature and uniformly positive spectral biRicci curvature.